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We consider the moduli space of bordered Riemann surfaces with boundary andmarked points. Such spaces appear in open-closed string theory, particularlywith respect to holomorphic curves with Lagrangian submanifolds. We consider acombinatorial framework to view the compactification of this space based on thepair-of-pants decomposition of the surface, relating it to the well-knownphenomenon of bubbling. Our main result classifies all such spaces that can berealized as convex polytopes. A new polytope is introduced based on truncationsof cubes, and its combinatorial and algebraic structures are related togeneralizations of associahedra and multiplihedra.

Deformations of bordered Riemann surfaces and associahedral polytopes